Optimal investment timing using Markov jump price processes

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O

PTIMAL INVESTMENT TIMING USING

M

ARKOV

JUMP PRICE PROCESSES

∗

Fernando A. C. C. Fontes

Depart. de Matem´atica para a Ciˆencia e Tecnologia,

Universidade do Minho

4800-058 Guimar˜aes, Portugal

Tel:351-253 510 400, fax:351-253 510 401 [email protected]

Dalila B. M. M. Fontes

Faculdade de Economia da Universidade do Porto

Rua Dr. Roberto Frias, 4200-464 Porto, Portugal

Tel:351-225 571 100, fax:351-225 505 050 Email:[email protected]

A

BSTRACT

In this work, we address an investment problem where the investment can either be made imme-diately or postponed to a later time, in the hope that market conditions become more favourable. In our case, uncertainty is introduced through market price. When the investment is undertaken, a fixed sunk cost must be paid and a series of cash flows are to be received. Therefore, we are faced with an irreversible investment. Real options analysis provides an adequate framework for this type of problems by recognizing these two characteristics, uncertainty and irreversibil-ity, explicitly. We describe algorithmic solutions for this type of problems by modelling market prices evolution by Markov jump processes.

Keywords: Irreversible investment, optimal stopping, dynamic programming, Markov jump

processes.

J.E.L. Classification. C61, G31.

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1 I

NTRODUCTION

The timing of market entry decisions is a central concern for business strategy. In a context of uncertainty and irreversibility the entry timing decision becomes even more important since by making a commitment we loose the option of waiting for a better opportunity.

The problem that we discuss here is deciding on the optimal timing for an irreversible invest-ment. A firm has an investment opportunity in the exploration of a natural resource, such as oil. When the investment decision is made, the firm pays a sunk fixed cost and from then on it will start to receive the incomes of the exploration over a long period, considered infinite. This prob-lem falls within the real options category, since at each point in time we have the option to invest or to wait for a more appropriate time and also because the investments discussed here typically involve real assets rather than financial assets (see e.g. (Dixit and Pindyck 1996 , Trigeorgis 1996 )).

The income received at each instant of time, the price of the resource, is considered to evolve according to a known stochastic dynamic model. Therefore, the expected total income will then be the integral over time of the expected discounted incomes received at each instant, i.e. the expected discounted prices of the resource integrated from the time of investment to infinity.

Usually, a simple static Net Present Value analysis of this problem is made. A comparison is made between the fixed cost of the investment and the total expected income. Questions like “Will the expected total income be greater then the investment cost?”, “If yes, what is the probability of that happening?”, and “How many years will it take for the investment to be paid?”, are typically answered by such analysis.

Here, we use a real options approach to this problem focusing on the optimal time for invest-ment. Even if the answers to the previous questions would lead us to invest, the expected evolution of the resource price could be such that the conditions in the future would be even better. In this scenario, although a decision to invest now is a profitable one, the decision to wait and invest in the future would be more valuable, as long as the opportunity to invest remains available. In the real options literature this is known as the option to delay.

The question addressed in this work is the instant of time at which the best conditions for investment occurs, i.e. the optimal investment time. To answer this, instead of comparing the present expected total income just with the sunk cost of investment, we have to compare the present expected total income with the expected total incomes of all future times and choose the maximum one. This problem, as we will see, falls within a category of problems known in the literature as “Optimal Stopping Problems” (see e.g. (Shiryayev 1978 , Peskir and Shiryayev 2006 )).

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specifies under which conditions we should invest. In fact, the results obtained state that the investment should be undertaken once the price reaches a certain threshold value. However, for a specific price value, it is possible to obtain an estimate of the waiting time till investment.

This problem can be addressed using a set of known methodologies to analyse and price Amer-ican options. The most used are the ones based on the Black-Scholes model ( Black and Scholes 1973) for infinite horizon problems (perpetual American options) with prices follow-ing geometric Brownian motion, as well as the discrete-time models such as the binomial and other lattice-based models (Cox, Ross, and Rubinstein 1979 ). In the latter set of methodolo-gies, most literature addresses the finite horizon case, since in opposition to the Black-Scholes based methods, these methodologies are more complicated in the infinite horizon case. Notable exceptions are the recent works of Boyarchenko and Levendorskiˇi (Boyarchenko and Leven-dorskiˇi 2007a, Boyarchenko and LevenLeven-dorskiˇi 2007b) proposing an elegant framework to ad-dress the infinite horizon case when prices follow binomial/trinomial lattices or a random walk.

Here we consider that the evolution of the price is modelled as a Markov jump process. The randomness of the process is considered to occur not in a continuous way, but rather at specific instants of time. This is particularly suitable when the main random factors that affect the price are point events, occurring at random times and with random intensity. There are several examples of price changes in natural resources caused by financial, political or environmental events. Oil prices are a typical example of a variable whose value is affected by abnormal news and their sudden variations are more adequately modelled by jump processes. A possible application for the model and algorithms here developed is the decision to make an investment in the exploration of an oil field.

In this work, we discuss 3 solution methodologies, leading to 3 different algorithms, for the optimal stopping problem modelling an irreversible investment decision when the prices follow a Markov jump process. For illustration purposes, though, we start by discussing a discrete time Markov model.

2 P

RELIMINARIES

Consider a Markov jump process {Xt; t ∈ R+} with transition function Pt, generator A, and

state-space E (countable). Let α be a nonnegative number. The α-potential of the function g (bounded, non-negative, and defined on E) for the process X is the expected value of the total discounted return of the rewards g, given by

Uα_{g(i) = E} i ·Z _∞ 0 e−αt_{g (X} t) dt ¸ ,

and the matrix Uαcan be computed as

Uα_{(i, j) =}

Z _∞ e−αt_P

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Proposition 2.1 Let g be a bounded function and α > 0 then the vector u = Uα·g is the unique solution to the system of linear equations

(αI − A) u = g.

Proposition 2.2 For any stopping time T

Uα_{g(i) = E} i ·Z _T 0 e−αt_{g (X} T)dt ¸ + Ei £ e−αT_Uα_{g (X} T) ¤ .

(See (C¸ inlar 1975 ) for the proof of these propositions.)

Let f be a finite-valued function defined on E, the state-space of a Markov process Xt with

semigroup Pt, and let r ≥ 0. Function f is said to be an r-excessive function for the process X

if for all x ∈ E and for all t ≥ 0

f (x) ≥ e−rt_P tf (x),

or equivalently

rf (x) − Af (x) ≥ 0.

A function which is 0-excessive is simply called excessive.

Theorem 2.3 The value function V is the minimal r-excessive function that majorises g.

(See (Shiryayev 1978 ) chapter 3, for a proof.)

Hence for a finite space E, V can be computed by linear programming as minV P j∈E V (j) s.t. (A − rI)V (x) ≤ 0 −V (x) ≤ −h(x) for all x ∈ E,

where the second set of constraints is active (satisfied as equality) when x belongs to the stop-ping set; otherwise the first set of constraints is active.

Hence, these constraints can also be written as the following Variational Inequality max {AV (x) − rV (x), h(x) − V (x)} = 0, ∀x ∈ E.

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3 D

ISCRETE

T

IME

M

ODEL

The problem here is to decide the best period of time to invest in the extraction of a resource given its actual price and a stochastic model of the price evolution. For simplicity of exposition, it is assumed that the investment can be done immediately, once decided, and the corresponding income starts on the next time period.

The prices are considered to evolve according to

pk+1 = (1 + wk)pk, (1)

where wk values are uncorrelated, belonging to a finite ordered set of values Ω={Ωx, ..., ΩN}

distributed according to F (Ωi)=Prob(wk ≤ Ωi) with corresponding mean Ewk = m and density

f (Ωi)=Prob(wk = Ωi).

Our goal is to determine the investment timing that maximises the net expected discounted profit

max k ( E " _∞ X i=k+1 (1 + r)−ipi) − (1 + r)−kI |p0, ..., pk #) ,

where I is the fixed cost of investment and r the interest rate.

Alternatively, we can write a dynamic programming recursion (see e.g (Bertsekas, D.P. 1976)) to decide at each time period whether to invest now or wait at least one more period.

max J0(p0) where Jk(pk) = max ( E " _∞ X i=k+1 (1 + r)−ipi) − (1 + r)−kI # , E(Jk+1(pk+1)) ) , (invest) (wait)

or using the value function at current prices

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Vk(pk) = max ( E " _∞ X i=1 (1 + r)−i_p i) − I # , (1 + r)−1_E(V k+1(pk+1)) ) . (invest) (wait)

For the infinite time horizon case, the case in which we are interested in, Vk(p) = V (p) for all

k, and so the value function V satisfies V (pk) = max ( Epk " _∞ X i=1 (1 + r)−ipi) − I # , (1 + r)−1Epk(V (pk+1)) ) , (invest) (wait)

which falls within the Optimal Stopping Problems category.

3.1 Solution method

Define h(p) as the net expected return of investing now, i.e. the sum of the discounted incomes from now till infinity net of the investment cost

h(pk) =E Ã _∞ X i=1 (1 + r)−ipi+k ! − I = ∞ X i=1 µ 1 + m 1 + r ¶_−i pk− I =1 + m r − mpk− I, if r > m

and υ(p) as the net expected return if we wait at least one unit of time

υ(pk) = (1 + r)−1E(Vk+1(pk+1)).

At time k, the decision is to invest if h(pk) > υ(pk), to wait if h(pk) < υ(pk), and either

decision is optimal if h(pk) = υ(pk) . If in this last case we choose to invest, the decision rule

is to invest if and only if h(pk) ≥ υ(pk), i.e.

iff 1 + m

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iff pk≥

r − m

1 + m(υ(pk) + I),

iff pk ≥ p∗where p∗ satisfies p∗ =

r − m 1 + m(υ(p

∗_{) + I).}

A graphical interpretation of the decision rule is given in Fig. 1.

Figure 1: Optimal stopping graphical interpretation.

Note that for p greater than p∗(inside the stopping region), h(p) is greater than υ(p) because we are in fact loosing an opportunity if we do not make the correct decision even if it is only for one instant of time. The value function V coincides with υ(p) for p less than or equal to p∗and coincides with h(p) for p greater than or equal to p∗.

To achieve the solution it remains only to compute υ(p).

3.1.1 Computation of υ(p)

We know that υ(pk) satisfies

υ(0) = 0, υ(p∗) = h(p∗),

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Let us consider separately the cases when (a)(1+w)p > p∗, and when (b) (1+w)p ≤ p*.

a) Case (1+w)p > p∗ ⇔ w > p∗_p − 1

Let N 1 be such that ΩN 1 = min {Ωi ∈ Ω : Ωi > p∗_p − 1}, i.e. N1 is the index of the first term

in Ω that satisfies the condition of being in this case.

υa_{(p) = (1 + r)}−1_{E{h[(1 + w)p]|w > Ω} N 1} = = (1 + r)−1 µ 1 + m r − mE[(1 + w)p|w > ΩN 1] − I ¶ = = (1 + r)−1 Ã 1 + m r − m P_N i=N 1(1 + Ωi)f (Ωi) P_N i=N 1f (Ωi) p − I ! .

And the probability of being in this case is

N X i=N 1 f (Ωi) = 1 − Prob µ w ≤ p∗ p − 1 ¶ = 1 − F µ p∗ p − 1 ¶ . b) Case (1 + w)p ≤ p∗ ⇔ w ≤ p∗_p − 1

Let N 2 be such that ΩN 2= max

n

Ωi ∈ Ω : Ωi ≤ p∗_p − 1

o

. i.e. N2 is the index of the last term

in Ω that satisfies the condition of being in this case.

υb_{(p) = (1 + r)}−1_{E{υ[(1 + w)p]|w ≤ Ω} N 1} = = (1 + r)−1 P_{N 2} i=1ν [(1 + Ω_P i)p] f (Ωi) N 2 i=1f (Ωi) .

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N 2 X i=1 f (Ωi) = Prob µ w ≤ p∗ p − 1 ¶ = F µ p∗ p − 1 ¶ .

Finally, using the Bayes rule, υ(p) is given by

(1 + r)υ(p) = · 1 − F µ p∗ p − 1 ¶¸ υa(p) + F µ p∗ p − 1 ¶ υb(p), (1 + r)υ(p) = I + m r − m N X i=N 1 (1 + Ωi)f (Ωi)p − I · 1 − F µ p∗ p − 1 ¶¸ + N 2 X i=1 υ [(1 + Ωi)p] f (Ωi).

3.1.2 Computing an estimate for the Optimal “waiting” time

Recall that the original problem was

max k ( E " _∞ X i=k+1 (1 + r)−ipi) − (1 + r)−kI|p0, ...,pk #) .

In the case that at the present time k, the decision obtained is to wait, we may wish to estimate (given the present data p0, ..., pk) for how long we should wait before investing. The answer is

given by

τ = min{t ≥ 0 : E[h(pk+t)|p0, ..., pk] ≥ E[υ(pk+t)|pk]} =

= min{t ≥ 0 : Epk[h(pk+t)] ≥ Epk[υ(pk+t)]},

which, as we have seen, is equivalent to

τ = min {t ≥ 0 : E[pk+t |p0, ..., pk] ≥ p∗}.

As E(pk+t|p0, ..., pk) = E(pk+t|pk)=(1+m)tpk, we get

τ = min {t ≥ 0 : (1 + m)t_p

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and so τ = min ½ t ≥ 0 : t ≥ ln (p ∗_/p k) ln (1 + m) ¾ .

Hence for each value of pk, the corresponding stopping time can be directly determined as

τ = ln (p

∗_/p k)

ln(1 + m).

3.1.3 Special Case

In the special case where the prices are monotonically increasing (i.e. Prob(w >0)=1) we have that F

³

p∗ p − 1

´

= 0 for p = p*, and so the expression for υ(p) simplifies to

υ(p∗_{) = (1 + r)}−1 ( 1 + m r − m N X i=N 1 (1 + Ωi)f (Ωi)p∗− I ) . Since υ(p∗) = h(p∗) and N P i=N 1 (1 + Ωi)f (Ωi)=1+m, we obtain (1 + r)−1 · (1 + m)2 r − m p ∗_{− I} ¸ = 1 + m r − mp ∗_{− I.}

Finally, after some algebra we get

p∗ = rI

1 + m. (2)

So p∗can be determined explicitly in closed form for this special case. In the general case, this equation can be used as an educated initial guess for p∗ in an iterative procedure.

3.2 Algorithm

An algorithm to compute the solution to this problem is the following:

1. Iteration index K=1

2. Set initial guess for p_*1 (e.g. by eq. (2))

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For p =0...p_*1 υ(p) = h(p∗1) p∗1 p, For p = p_*1...P_max υ(p) = h(p). 4. Update estimate of υ(p) For p=0..P_max υ(p) = (1 + r)−1 ( 1 + m r − m N X i=N 1 (1 + Ωi)f (Ωi)p − I · 1 − F µ p∗ p − I ¶¸ + + N 2 X i=1 υ [(1 + Ωi)p] f (Ωi) ) . 5. Stopping condition If max

p |υ(p) − υOLD(p)| < ε then STOP.

6. Update estimate of p∗

p∗k+1_{= min {p : υ(p) = h(p)}.}

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3.3 Alternative Solution Methods

3.3.1 Reformulating as a finite state Markov chain model

As the evolution of the prices considered before is a process with independent increments, it satisfies the Markov property. In order to get a finite state-space (the previous case had not a finite state-space) we first apply logarithms to both sides of equation (1) getting

log Pk+1 = log Pk+ log (1 + wk) .

Define

Xk = log Pk,

we have now a countable state-space for the process {Xk, k ∈ N}. By setting upper and lower

bounds for the state-space, Xmaxand Xmin, which we define as x1 = Xmin and xn = Xmax, we

get a finite state-space X = {x1, x2, ..., xn} . Naturally, by clipping the state-space in this way,

we get a different process, but in a real problem application, the majority of the possible prices will certainly be within a bounded interval with high probability and so the difference will not be meaningful.

This being case, the new gain function h(x) is defined as

h(x) = E " _∞ X i=1 (1 + r)−i_eXi # − I,

which is the (1 + r)-potential of the exponential function minus the investment cost I.

Defining

h = [h (x1) , ..., h (xn)]T ,

f = [exp (x1) , ..., exp (xn)]T ,

I = I ∗ [1, 1, ..., 1]T,

and Inas the identity matrix of dimension n.

By Proposition 2.1

h = (In− (1 + r)Q) f − I.

Thus, the problem

V (X0) = max k E h (1 + r)−kh (Xk) i ,

can be solved for this finite state-space case, simply by calculating V for all x ∈ X, such that

V satisfies V (x) = max {h(x), T V (x)} , where T V (Xk) = (1 + r)−1EV (Xk+1) = (1 + r)−1 X xj∈X Q (Xk, xj) V (xj).

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Alternative Solution Method 1 The function V can be computed iteratively as ½

V0(x) = h(x)

Vm+1(x) = max {h(x), T Vm(x)} ,

and Vm → V as m → ∞.

Alternative Solution Method 2 Knowing that the value function V is the minimum (1 + r)-excessive function that majorises h, it can be computed by linear programming as

min P x∈X V (x) s.t. V (x) ≥ h(x) V (x) ≥ (1 + r)−1_{T V (x)} V (x) ≥ 0.

These alternative solution methods are adapted and implemented for the case of the Markov jump processes, as developed in the next section.

4 M

ARKOV

J

UMP

M

ODEL

In this section, we consider the price to be modelled as a Markov jump process. Such process is able to capture price jumps that are induced by the occurrence of rare events typically related to the arrival of new information. This information might be of different nature: technological (innovations)(Greenwood, Hercowitz , and Krussell 1997 , Galor Tsiddon 1997 , Grenadier and Weiss 1997 ), competition (new product or competitor entry)(Ghemawat Kennedy 1999 , Bresnahan and Greenstein 1999), political (wars, expropriation, change in legislation) (Clark 1997 , Wagner 1997 ), natural conditions (natural hazards, hurricanes)or even various types of rare events simultaneously (Martzoukosand Trigeorgis 2002 ).

Consider the resource price ptto evolve according to

pt = eXt,

where Xt is a piecewise constant Markov process, taking values in a finite ordered set X =

{x1, x2, ..., xn}, having a jump rate λ and with the post-jump location defined by the transition

probabilities Q(i, j) = Prob (XT = xj|XT = xi) in which Q(i, i) = 0 for i=0...N.

For a constant jump rate λ the transition function is given by

Pt= ∞

X

Qne−λt(λt)n

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If the jump rate λ is dependent on the state, the transition function is given by Pt(i, j) = e−λ(i)tδij + R_t 0 λ(i)e−λ(i)s P k∈E Q(i, k)Pt−s(k, j)ds, where δij = ½ 0 if i 6= j, 1 if i = j.

The generator A for this process is given by

A(i, j) = ½

−λ(i) if i = j,

λ(i)Q(i, j) if i 6= j,

or in matrix notation A = Λ(Q − I) where Λ = diag(λ(1), λ(1), . . . , λ(N )).

Our problem is to decide the time instant in which to invest that maximises the net expected discounted profit ρ(x) = sup τ ≥0Ex £ e−rτh(Xτ) ¤ ,

where h is the expected income at the time of investment.

4.1 Solution Method

In order to determine the optimal stopping policy, we first compute the gain function h, the net expected income at time of investment. Define

f (Xt) = exp(Xt)

and V (x), the r-potential of f , as

V (x) = Ur_{f (x) = E} x ·Z _∞ 0 e−rt_{f (X} t)dt ¸ .

The vector V=[V (x1), V (x2),..., V (xN)]T can be computed as the unique solution of the system

of equations

(rIN − A)V = f

where f =[f (x1), f(x2),..., f(xN)]T and IN is the identity matrix. Hence, the vector h=[h(x1),

h(x2),..., h(xN)]T is given by

h = V − I,

where I=I*[1,1,...,1]T.

To compute the value function we develop, for the Markov jump processes case, the three methods referred to previously corresponding to the following3 different algorithms:

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1. By using dynamic programming arguments, we can compute an approximation of the value function when the inter-decision time δ is a “small” value.

2. By using known properties of the value function we can reformulate the problem as a linear programming one.

3. By using dynamic programming arguments, the value function can be computed by analysing an increasing sequence of allowed jumps.

4.1.1 Solution Method 1

This method follows closely the first method described in the previous section and it is based on determining an approximation of the function ρ.

Defining

νδ(xt) = e−rδρ(xt+δ), (3)

as the net expected return if we wait a small time δ, we have the obvious result lim

δ ↓0νδ(x) → ρ(x).

This means that, for very small δ, νδgives us a good estimate of ρ, and so an “almost optimal”

solution can be achieved by applying the following rule

Invest iff h(Xt) ≥νδ(Xt), iff Xt≥ Xδ∗, where X_δ∗ is given by X∗ δ = inf {x ∈ X : h(x) ≥ νδ(x)};

Otherwise, wait for a time τδ, given by

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Now, we develop νδ, knowing that it is given by

νδ(Xt) = e−rδρ(Xt) = e−rδE max {h(Xt+δ), νδ(Xt+δ)}.

Similar to what we have done for the discrete time case, let us consider separately the two cases when a) Xt+δ ≥ Xδ∗ and when b) Xt+δ < Xδ∗.

a) Case Xt+δ ≥ Xδ∗

We start by defining Pa, the probability of being in this case as

Pa= P rob(Xt+δ ≥ Xδ∗ |Xt = xi) = N

X

j=1

p(i, j, δ)Ixj≥X∗.

Since a Poisson process for small intervals [t,t+δ[ can be given by

Prob(Nδ = m) =    λδ + o(δ) if m = 1 o(δ) if m > 1 1 − λδ + o(δ) if m = 0 , (4) Pacan be rewritten as Pa= λδ N X j=1 qijIxj≥X∗+ (1−λδ)Ixj≥X∗. (5) And so νa δ(Xt) = e−rδE max {h(Xt+δ)|Xt+δ ≥ Xδ∗} = = e−rδ λδPN j=1 h(xj)qijIxj≥X∗+ h(xi)(1−λδ)Ixi≥X∗ Pa . b) Case Xt+δ < Xδ∗

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Pb=Prob(Xt+δ < Xδ∗ | Xt= xi)= = λδ N X j=1 qijIxj<X∗+ (1−λδ)Ixj<X∗, (6) we have νb δ(Xt) = e−rδE max {ν(Xt+δ)|Xt+δ < Xδ∗} = = e−rδ λδPN j=1 ν(xj)qijIxj<X∗ + ν(xi)(1−λδ)Ixi<X∗ Pb

Finally νδ(x) is then given by

νδ(x) = gνδa(Xt)Pa+ νδb(Xt)Pb = = e−rδ ( λδ N X j=1 qij[h(xj)Ixj≥X∗+ ν(xj)Ixj<X∗] + (1−λδ)[h(xi)Ixi≥X∗+ ν(xi)Ixi<X∗] ) . (7) Having a recursive equation for νδ(x) we only need a fixed point to start with.

Choosing this point to be the largest value in X, say xN, we get

νδ(xN) = h(xN).

If this is not true for this point, then it is also not true for any other, and then we would have the trivial solution “never to invest”.

Algorithm

1. Determine h(x)

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3. Initialise ν_δ(x) as ν_δ(x) = h(x)

4. Update estimate of νδ(x) for i = 1, ..., N

½

νδ(XN) = h (XN)

νδ(xi) = ...(by eq. (7))

5. Stopping condition

If max

i |ν(xi) − νold(xi) < ε| then stop

6. Update estimate of X∗

X∗ _{= min}

i {xi : ν(xi) ≤ h(xi)}

7. Goto 4.

The previous method can be supported by the known result that the value function is the minimal

r-excessive function that majorises h; where the r-excessive function is defined as a function f satisfying:

f (x) ≥ e−rt_P tf (x)

or

rf (x) − Af (x) ≥ 0.

We no longer have to consider any value δ for a discretization of t, because we are already using an infinitesimal operator - the generator A. It follows the result of the Variational Inequalities

max {rV (x) − AV (x), h(x) − V (x)} = 0, ∀x ∈ E, which can be formulated as a linear programming problem

min P j∈E V (j) s.t. AV (x) − rV (x) ≤ 0 V (x) − h(x) ≥ 0 for all x ∈ E,

An implementation of this method in MATLAB was carried out, where a MATLAB built in function for solving the linear programming problem was used.

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Another particularity of this problem is that since r >0 and Xtis constant between jumps, the

optimal solution is always be to invest immediately after a jump.

This happens since if for a certain value Xtthe decision is to invest, we should do so as soon as

possible, as we would only be loosing money by waiting.

This can be justified with an example.

Consider Xt= x constant, then

h(x) = Z _∞ 0 e−rt+xdt − I = e x r − I, and so ρ(x) = Sup τ ½ e−rτ µ ex r − I ¶¾ , gives, obviously, τ =0 if r >0.

Therefore, instead of considering a small time instant δ as in the previous method, we can consider Tn, the time of next jump.

Define the best value obtainable considering up to n jumps as

Jn(xi) = Sup τ ≤Tn Ei £ e−rτh (Xτ) ¤ .

It can also be given, as we have seen, by

(keep strategy) (invest at the n-th jump)

Since Ei £ e−rT1_{h (X} T1) ¤ = ∞ Z 0 λe−λt_{dt ·} N X j=1 Q(i, j)h(xj) = λ λ + r(Qh)i,

Jn(xi) can be computed for i=1...N and for n=1,2,... as

lim n→∞Tn= ∞, and so Jn(x) −→ n→∞_{τ <∞}SupEx £ e−rεh (Xτ) ¤ = ρ(x),

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Algorithm

1. Determine h (as in the previous methods)

2. n=0, MinIter=10 FOR i=1...N J0(xi) = h(xi) 3. REPEAT δmax=0 FOR i=1...N Jn+1(xi) = max ( Jn(xi), µ λ λ + r ¶_n+1 ¡ Qn+1_h¢ i ) −δ = |Jn+1(xi) − Jn(xi)|

δmax = max {δ, δmax}

n = n+1

UNTIL δ_max <ε ∧ n ≥ MinIter

4.2 Example and comparison of the algorithms

To illustrate the algorithms consider the following example:

Interest rate: r=1,

Fixed cost of inv.: I=10,

XM ax = ln(200),

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X = {XM in, ..., XM ax} with N=20 points equally spaced,

Jump rate λ=10.

For the 1st Algorithm (corresponding to solution method 1):

Error ε=1e-3.

δ=1e-4.

For the 3rd Algorithm:

Error ε=1e-8,

Results of the 1st algorithm

Xstar=X10=1.817

Iteration:1 Error:1.059 Xstar=X13=2.504

Final Result:

This algorithm converges very fast if δ<<ε. Moreover, reducing δs the difference between νδ

and h for X >= X∗ is reduced. However, if we reduce δ too much, we start having numerical problems (because the term that is multiplied by δ vanishes).

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Figure 2: 1st algorithm results.

Xstar=X15=3.102

Figure 3: 2nd algorithm results.

This algorithm was the fastest to run for this example, but we have used a MATLAB built-in function to determbuilt-ine the optimal solution of the lbuilt-inear programmbuilt-ing problem. Although linear programming optimisation methods are usually very efficient (the Simplex algorithm, for example, usually converges in few iterations), this efficiency is not guaranteed for every instance of the problem.

Results of the 3rd algorithm

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Iteration:10 Error:0.002785

Iteration:20 Error:0.00002414

Iteration:30 Error:2.093e-007

Iteration:37 Error:7.539e-009

Xstar=X15=3.102

Figure 4: 3rd algorithm results.

As expected we have found the same solution as with the previous methods. Conceptually this method is, probably, the most simple, resulting in a more clear implementation. Therefore, it can be more easily expanded to cover more general Markov processes or additional problem features. In fact, we are developing this solution method for Markov processes that in addition to random jumps allow a deterministic drift – known as piecewise deterministic Markov processes (Davis 1993 ).

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